Still logged in?

I'm using Cookie 6.0.14 with Safari 13.1. When I manually choose Remove Safari Unwanted Data, if I go back to a site where I was logged in before (e.g. this forum, Amazon, whatever), it still recognizes me as logged in. This does not happen if I go to Privacy/Manage Website Data/Remove All in Safari's preferences; after that, sites will not recognize me as still logged in.